Ab initiocharacterization of the quantum linear-zigzag transition using density matrix renormalization group calculations

Abstract

Ions of the same charge inside confining potentials can form crystalline structures which can be controlled by means of the ions density and of the external trap parameters. In particular, a linear chain of trapped ions exhibits a transition to a zigzag equilibrium configuration, which is controlled by the strength of the transverse confinement. Studying this phase transition in the quantum regime is a challenging problem, even when employing numerical methods to simulate microscopically quantum many-body systems. Here we present a compact analytical treatment to map the original long-range problem into a short-range quantum field theory on a lattice. We provide a complete numerical architecture, based on Density Matrix Renormalization Group, to address the effective quantum φ 4 model. This technique is instrumental in giving a complete characterization of the phase diagram, as well as pinpoint the universality class of the criticality.

Finite-density phase diagram of a(1+1)−dnon-abelian lattice gauge theory with tensor networks

“…− f M 0 acquiring a finite nonzero value at the thermodynamical limit, for some k (precisely: k = π for f M = 1, while k = 2π/3 for f M = 2/3). To avoid potential problems of symmetry restoration due to finite-size simulations, we extract the CDW order parameter from the static structure factor, at wave-vector k, of the matter density-density correlations [66,67] Fig. 3 we show the typical behaviour of the relevant order parameter ζ k , at filling f M = 1 (f M = 2…”

The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

“…In this representation, the motional modes of each particle are often truncated to a compact dimension by selecting d orbitals {|i } to act as local canonical basis, e.g. those with lowest local energy, or the solutions of a mean field approach [109,110].…”

Operator-based derivation of phonon modes and characterization of correlations for trapped ions at zero and finite temperature

“…In this work we will not include the effect of the quantum fluctuations on ∆, although the formalism provides a systematic starting point to consider them within a Hartree-Fock Bogoliubov-de Gennes approach. Both the equilibrium value of ∆, as well as the transition point κ c are only modified slightly by the full quantum theory [19,20], as relevance of the quantum corrections is determined by the ratio of the local harmonic oscillator length scale and the distance between the ions d, which is very small in current experiments.…”

“…At sufficiently low temperatures, and depending on the interactions, geometry and parameters of the trap, the contained ions form a variety of crystalline structures [1][2][3][4][5][6][7][8][9][10][11]. Particular focus has been cast upon the transition between the linear and the zigzag configurations for a long 1D ion chain, both experimentally [1][2][3][4] and theoretically [12][13][14][15][16][17][18][19][20][21].…”